This looks a little tricky. The assumption that would be used in order to just add the probabilites is that the Events are Mutually Exclusive rather than independent. The probability of independent events could be added if they were mutually exclusive. But in fact, barring the trivial case, if they are independent they can't be mutually exclusive. In this case the Events are not independent so they might be mutually exclusive. But if we do consider them mutually exclusive, thus allowing us to just add the probabilities, then we have the problem you are pointing out. Your point being the same then, that we should not treat the two bastards (or {T-monday},{T-tuesday} ) as two mutually exclusive events but as one atomic outcome, "writes bastard twice", or with SB "awakens two days".

btw, I think you're making excellent arguments. But if you apply the indifference principle to what day it is when Tails to get the (.5,.25,.25) model, how do you deal with the conditional probability this model produces given Monday, ie P(Heads|monday) = 2/3 ?


PairTheBoard

Yeah it's {0.5, 0.5} model not {0.5, 0.25, 0.25}. You can't really apply indifference principle here, as indifference principle applies to events which are mutually exclusive and as MT and TT are the same event they don't satisfy this requirement.

You can try to introduce 2nd coin and define your space S with the following states: HH, TT, HT, TH. With events being:

1) {HH U HT}
2) {TT}
3) {TH}

This model describe SB experiment if she is only asked once during whole experiment. As she is always asked, the events are really:

1){HH U HT}
2){TT U TH}
3){TH U TT}

so 2) = 3) and asking in which exactly SB is is the same as asking to decide if 4 is in {4,5,6} or if 4 is in {6,5,4}.

P(H|monday) = P(the accurate description will be the heads path|a monday awakening is included in the path), and the latter is P=1 (just like being awakened at all) for H, T, and overall. You can modify the propositions to something like "awakened with amnesia, asked credence, told day, asked credence again, put back to sleep." and it becomes obviouly uninformative. monday isn't being picked from a day or an (HM/TM/TT) awakening distribution

You believed a monday awakening was coming with P=1 regardless of the coin. Now it came. Your credence for the accurate description of the past 4 days, come wednesday, clearly can't change.

It's 1500. You can have more results than you have trials if a trial can produce two results.

Pretend there's no memory erasure and SB can watch the flip. You clearly have 1000 coin flips, 1500 awakenings, and 500 times SB is 100% H, and the other 1000 times SB is 100% tails.

Now same thing without observing flip. 500 times she is 100% sure it's tails (cuz she remembers already being woken up, therefore it's TT).

The other 1000 times she says 50/50.

There are clearly 1500 points that need to be included in the experiment, not 1000.

.

Yes, of course you can have more results but what the definition calls for are trials (or more specifically subset of trial which satisfy your condition).

Read the definition man. It's nx/nt where nt is number of trials, not "points".
Again consider my dice rolling experiment to see where your thinking leads to: we roll a dice and events are:
a)roll in {1,2,3}
b)roll in {4,5,6}
c)roll in {6,5,4}

After every event we check if a), b) c) are satisfied and write "X" for every satisfied event.
After 1000 trials there will be 1500 "points". To determine probability of event a) you calculate: (number of trials where a) was satisfied) / (number of all trials) which is 500/1000 not 500/1500. "Points" don't matter.

"Points" are not "included in the experiments" they are merely outcomes of it. They are events which were satisfied. You don't use them to determine probability. There is no place in the definition of probability which include those "points" in a denominator.

The only thing which is "clear" is that you refuse to apply definition of probability.
Focus on it: "nx / nt". Number of TRIALS in denominator not "points", not "events", not "results". Trials.

I've done it twice already ITT even though I think it's unnecessary.

http://en.wikipedia.org/wiki/Frequency_probability

We are debating the value to assign to the variables in any definition, which is why the definition isn't important.

If you want to use P(x)=nx/nt

You could argue that each awakening is a trial which would give a clear 500/1500.

You could also argue that formula basically applies to the flip = 1/2, but then you would have to say, "Given the probability of the coin is 1/2, but trials happen at a ratio of 2:1 T:M, when placed in an interview..."

You get the point.


What you CAN'T do, is define a trial as a flip, and then COMPLETELY IGNORE the ratio of awakenings TO ANSWER A QUESTION BASED ON AWAKENINGS.



There are plenty of ways to frame the problem, but you can't mix and match strategies that look at the odds of the flip itself, and conclude what SB determines with information that you never included in the math.

You edited your post after I had already responded...

An interview certainly sounds look a trial to me.

Each interview is pretty clearly a trial here.

If you think the question is, "What are the odds of a coin", you will disagree with me, but that is not the question.

The question is, "When interviewed, what does she answer?"

When X, what does she answer - X certainly seems like it might be a trial.

Of course the real answer is there are multiple trials going on in this experiment, but it's really unnecessary to do the math that way.

No
How do you arrive at 500 in the first place ?
You do it by running the experiment 1000 times, right ? Trial is defined as run of the experiment.

No, awakenings are results of the experiment. Trial is a run of experiment.

Trials doesn't happen at ratio. Trials are runs of the experiment. You decide to run the experiment 1000 times then you are fixed with 1000 trials for all subsequent discussion.
Only if asked about conditional probability you could consider subset of this 1000 trials (those which resulted in condition of your choice), never super set of it.

Man, the very definition of a trial is one run of a experiment.

Btw, what is your answer to SB problem with only one awakening during the experiment ? If we can't get anywhere with definition of probability at least answer that one.

Dude, it doesn't matter. What if SB is explicitly told the outcome of the coin? Are you going to do the same exact math to tell me "50/50, see formula".

There's more information. It's not 50/50. Your formula is not relevant if it doesn't factor in all the relevant information.

If there's no TT? Clearly 1/2 is the answer.

If SB is told the outcome of the coin, let's say "heads" she will do the following math:
There were 1000 experiments. In 500 of them the coin was heads and I am clearly in that 500. I now apply definition of probability taken from wikipedia which says:
"nx/nt". What is nx ? Clearly 500 (becaue out of 500 trials I am now concerned about in 500 of them the coin was heads). What is nt ? Again 500 (because out of 1000 overall trials I am told to only worry about the ones where the coin was heads) so 500/500 = 1.
Not surprisingly by applying the definition of probability we arrive at the conclusion that if SB is told the coin is heads then her credence about the coin being heads is 100%.

I mean the experiment as in original SB she is still awakened two times if the coin is tails but only asked about her credence once per experiment.

Edit: Just took the same post and put it inside the spoiler because #263 is a more immediate response to post #261, and this might just be a very unnecessary post. Feel free to ignore this post.

Ok. This is the step you have left out of the previous formula then.

"There were 1000 experiments, it is twice as likely I am in one of the 500 experiments where I am awoken twice"

Put that into the formula.

Answer clearly depends on the exact phrasing of the question. If on TT she is awoken and never interviewed, but MT and MH happen normally, then the answer is, "She will think 1/3 upon awakening, and 1/2 upon being interviewed. There is clearly information being added when she is interviewed."

What was the process which puts me in a room ? Assuming it was random symmetrical process I am in every room with 1/3 probability. This is big assumption which wasn't spelled out in description and is necessary to answer this question.
We now run the experiment 1200 times. First we assign me to a room and then we flip a coin. There are following events:
a) (me in H room)&(coin heads)
b) (me in H room)&(coin tails)
c) (me in T1 room)&(coin heads)
d) (me in T1 room)&(coin tails)
e) (me in T2 room)&(coin heads)
f) (me in T2 room)&(coin tails)

As all of them are mutually exclusive I can happily assume p(a) = p(b) = p(c) = p(d) = p(e) = p(f) = 200/1200 = 1/6
Now I am asked about the coin. This means I am only concerned about trials which resulted in a), d) or f) which is overall 600 trials. Out of those trials in how many of them the coin is heads ? By looking at the table above I can see that it's 200 trials (the ones which resulted in a))
My answer is therefore nx/nt or 200/600 or 1/3.

Note that my answer will change if I start having some doubts about the process which was used to choose a room for me. For example if I somehow get to know that scientist running this experiment likes to put his subjects in M room and only rarely in T1/T2 rooms I need to reconsider my calculations.

No. Every trial is as "likely" as the other. You just count number of trials out of all trials. You don't assign weight to them.
I mean, can you spell your argument for 1/3 again using probability space, events and trials ?
So far you were using 50/150 somehow but I hope you can now see that you either have "50" with 100 trials or "150" with some other number of satisfactionary trials. There is no way 50 and 150 go together.

BUT YOU JUST DID IN POST 261!!!!!

No, I didn't. You are so fixated on your misunderstanding of definition of probability that you just can't see above it.
I've chosen to run the experiment 1000 times in post #261. Those are 1000 possible universes if you will. You then put the condition on me "I tell you that the coin is heads". I now review my 1000 universes and see in how many of them I could hear such a statement. I can count 500 of those and this is my "nt" in denominator from wikipedia article.
Now you ask me "assuming I told you the coin was head, how often do you think the coin is head". I did "assuming" part already and came out with 500 interesting universes. Now I review those 500 universes and count in how many of them the coin is head. If you didn't lie I come up with "every one of them, so 500".
Now using my favorite wikipedia article with neverworn out formula: "nx / nt" I arrive at 500/500 = 1.

You are assigning weight to the trials. Some you give 100%. Some you give 0%.

How is this not assigning weight?

You absolutely can and should assign weight.

Going to sleep. Think for yourself, "Am I approaching these two problems the same way, or am I adding a step in one and ignoring the similar added information in the other?"

You should be able to identify this inconsistency if you reflect on it for a while.

Man, definition. Jesus. Please read it:
If we denote:
E - some event
A - some other event
x - set of trials which satisfy E out of all trials
t - set of all trials
n(y) - number of elements in set y

then:

(1) p(E) = n(x) / n(t)

(2) p(E | A) = n(Ax) / n(At) where Ax is subset of x made of elements of x which satisfy A and At is subset of t made of elements which satisfy A.


Now in the experiment when you told me what the coin was:
E - the coin is head
A - I heard from you that the coin is head
x - those trials out of all trials which satisfy E (where coin is head); n(x) = 500
t - all the trials (so 1000 of them)

Applying (1):
p(E) = n(x)/n(t) = 500/1000 = 1/2
This describe my credence before you say anything.

Applying (2):

p(E | I heard the coin was heads) = p(E | A) = n(trials from x where I heard "the coin is heads") / n(trials from t, where I've heard "the coin is heads") = 500/500 = 1.

In SB problems it's straightforward as well:

E1 - SB awakened on MH
E2 - SB awakened on MT
E3 - SB awakened on TT
A - SB asked about her credence
xy - those trials out of all trials which satisfy Ey (so 500 of them for every y)
t - all the trials (so 1000 of them)

Now apply the definition and see what it yields.

I feel that I hit the wall in discussion with you as you just refuse to use definition of probability and you keep coming out with new meaning for "trials", "results" and new definition of probability.


Yeah, same for you. I am just directly applying the definition. I am not adding any weights, I am not making up ratios like (results I like) / (all results) and call them probability. I am just counting trials and see which ones satisfy which events (any event is either satisfied or not satisfied by the run of the experiment, it can't be "satisfied with 50% weight" or anything like that).
You somehow want to run the experiment 100 times and then use number 150 to determine probability which is obviously wrong as you can only use 100 or some number less than 100 (in case of conditional probability).

I even drew nice picture for you which let's you see how definition is applied to conclude p(MH) = 1/2.
The only thing I can do now is to wait for your description of probability space which describes SB experiment and for you to apply the definition of probability, not some random ratio of outcomes to determine the answer.

ok.


PairTheBoard

1.Coin is flipped 1000 times.
2.Sleeping beauty is woken up 1500 times (500 times for heads on monday, 500 times for tales monday, 500 times for tales tuesday).
3. The coin being flipped is not the trial! The trial is the awakening based on the coin flip, not the coin flip itself!

If sleeping beauty is woken up, 2/3 of the time it is because of the tail flip. It doesn't matter that the probability of a tail being flipped is only .5.
If sleeping Beauty were woken up not only on Monday and Tuesday but also Wednesday from a tail flip the probability that any awakening was caused by heads falls to to 1in3. If she is woken up N days for Tails the probability that any awakening is from heads is 1/(1+N).

Coin is flipped 1000 times.

One blue ball is put into the bucket if the coin was heads and two green balls are put into the bucket if the coin was tails. So after overall 1500 balls are in the bucket after 1000 flips.

The coin being flipped is not the trial ! The trial is taking random ball form the bucket (which is based on the flip) not the flip itself !

If we hold a ball in our hand without looking at it, 2 times out of 3 it's the ball put into the bucket because of tail flip.

It doesn't matter that the probability of a tail being flipped is only .5

So now if we pull random ball from the bucket and are asked about our credence about the coin flip we should answer 2/3 for tails and 1/(1+N) in general case.

This is of course wrong and if you disagree I invite you to play this game for money.
It also shows why you need to describe experiment you are performing and probability space which models it.
What you are doing, ZJ is doing and most 1/3'ers are doing is mixing terms. You first flip coin 1000 times then you use it to describe completely new experiment but use amount of times you performed the old one to get data in your new one. This is total confusion.

If you want to make credible argument you need to describe what experiment you are performing, what's the value you seek to determine, what are events in your space etc.
The moment you say "I flip the coin 1000 times and now I have 1500 trials" we can safely conclude you are engaged in some black magic which has nothing to do with probability or mathematics.

I feel I hit the wall already in this thread so I will try to refrain from indulging in it anymore.
Just realize that next time you perform something 1000 times then call outcomes of this exercise "trials" and then use ratio of number of elements in subset of those outcomes to number of all your outcomes then call it probability and if nobody responds to it, it's not because you are right but because it's hopeless task.

You are so close. If you pull out a ball without looking at its color and are asked your credence for whether the ball was put in on a heads, what should you answer? When you awaken in SB and are asked your credence for your being awakened on a heads, what should you answer?

Seems like the only people that are wrong in this thread are the ones picking a side. Or the ones that are redefining things (slightly, almost imperceptibly, but enough to completely change the question)

This is wrong.